Most powerful tests are used when both the null and alternative hypotheses are simple. This means that the parameter takes a single value under both the null and alternative hypotheses.
What is a most powerful test?
Suppose that you want to conduct a hypothesis test for the value of a parameter θ with simple point hypothesis given as
H0: θ=θ0 vs H1: θ=θ1
Out of the many possible critical regions with size α, the test having the critical region with the greatest power among all of the others is called the most powerful test.
Generalization: The generalization of the concept of most powerful tests is UMP (Uniformly most powerful test) which is used when the alternative hypothesis is a composite hypothesis. In a composite hypothesis, the parameter is allowed to take a range of values.
Most Powerful tests and Likelihood Ratio:
It is known as a consequence of the Neyman Pearson lemma that the likelihood function gives us the most powerful tests when both hypotheses are simple. By using the likelihood ratio as our test statistic we can decide whether to accept or reject the null hypothesis.
Formal Mathematical Definition of Most Powerful Tests:
